Production Networks

Kenan Huremovic∗ Fernando Vega-Redondo†

First version: October 2014This version: September 2016

Abstract

In this paper, we model the economy as a production network of compet-itive firms that interact in a general-equilibrium setup. We start by charac-terizing how the structure of production network determines the profit of afirm as well as the social welfare. Then we proceed to conduct a range ofcomparative-static analyses on the effect of both various types of price dis-tortions and changes in the network structure. We discover that their impactis in general nonmonotone, depends on global network features, and affectseach sector depending on the overall pattern of firm centralities. We alsoshow that the inter-sector linkages underlying these effects can usually bedecomposed into a push and a pull component, in the spirit of Hirschman(1958). Finally, we illustrate that if policies of support and shock mitigationare evaluated from a dynamic viewpoint, the reliance on strict market-basedcriteria can be quite misleading in terms of social welfare.

JEL: C67, D51, D85Key words: Production, Networks, Distortions, Centrality, Profit

∗Aix-Marseille University (Aix-Marseille School of Economics), CNRS, & EHESS, E-mail:

kenan.huremovic@univ-amu.fr†Bocconi University & IGIER, Email: fernando.vega@unibocconi.it

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1 Introduction

Any modern economy is a complex network of interfirm buyer-seller relationships

that constitute its production structure. There is a growing interest among econo-

mists in adopting this network perspective to study a wide variety of economic

phenomena: trade, intermediation, innovation, technological diffusion, learning, or

the transmission of shocks. This, of course, has prominent precursors in the cele-

brated work of John von Neumann, Wassily Leontief or Albert O. Hirschman, all

of whom stressed the importance of interindustry relationships (or linkages) for a

proper understanding of some of the key characteristics of an economic system.

In this tradition, the present paper proposes a general-equilibrium model of the

economy that, despite being standard in almost every respect, highlights the net-

work of interfirm relationships on its production side. In a nutshell, our objective

is to obtain a precise understanding of how the details of the production network

topology (on which no a priori restrictions are imposed) shape profits, welfare, and

the effects (static and dynamic) induced by a wide variety of changes and policy

interventions.

In order to focus our analysis on the production structure of the economy, the

demand side is modeled through a representative consumer while the technology of

each firm is assumed to be of the Cobb-Douglas type. In general, each firm uses a

diverse range of inputs, whose productivities are individually specific. Under these

conditions, our analysis starts by showing that a unique equilibrium exists that

displays a very sharp relationship to the network structure of the economy. Specifi-

cally, we find that, at equilibrium, the profitability of any given firm is proportional

to its network centrality, as given by a suitable generalization of the well-known

measure of centrality proposed by Bonacich (1987)1. On the other hand, we also

determine the consumer’s welfare at equilibrium, providing closed expressions for

how it depends on the different parameters of the model (in particular, the density

and symmetry of the production structure).

Next, the paper turns to studying how the equilibrium is affected by a wide

variety of different changes in the environment. First, we consider the impact of

1For example, as compared to the received notion of Bonacich centrality that considers thepaths of different lengths that join any given pair of nodes, in our case only those that connecta firm node to a consumption node are considered. The weight attributed to each of these pathsreflects not only its length (as in the standard Bonacich centrality, they are discounted by theirrespective length) but also the relative importance that the consumer’s utility function attributesto the end good in question (while in the standard notion all end nodes are given a uniformweight).

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distortions, formulated in the idealized form of price wedges – either general ones

that uniformly affect the whole economy, or individual ones applying to single firms.

Then, we turn our attention to structural changes in the production network and

consider both the effect of global transformations that affect either connectivity

or the number of nodes, as well as local changes that affect only single nodes or

individual links. We then close this part of our analysis with a comparative study

of supply-channeled changes (such as technological improvements) with those that

operate through variations in demand (e.g. a rise or fall in the consumer’s relative

preference for a particular consumption good).

For all of the cases listed above, we provide formal expressions that capture the

total impact of the change in an explicit form. These expressions are conceptually

quite simple and should prove helpful in empirically evaluating alternative measures

of economic policy. Furthermore, many of them can be understood in a quite parallel

fashion, in that they involve a similar integration of constituent effects. Thus, on

the one hand, they can be regarded as consisting of a first-order impact on the

immediate agent experiencing the change (e.g. the firm subject to an individual

tax or subsidy), followed by a spread of such a first-order impact throughout the

whole network as captured by a suitable matrix of incentralities. On the other

hand, the impact of many of those changes can also be decomposed in terms of

a push effect that operates downstream and a pull effect that does so upstream.

This dichotomy is reminiscent of the distinction between forward and backward

linkages often considered in policy analysis and notably proposed by Hirschman

(1958). Here we show how to distinguish precisely between them and also compare

their implications: while forward linkages induce resource reallocation downstream

but have no effect on profits, revenues, or input demands because of the entailed

adjustment of prices, backward linkages significantly alter all those variables – i.e,

not only prices but also revenues, profits, and input demands.

The analysis advanced so far is inherently static, i.e. it compares how different

changes in the parameters of the model affect equilibrium magnitudes. By building

on it, however, we can also address some genuinely dynamic issues. In this paper,

we simply outline the problem, leaving for future work an exhaustive study of it. We

consider, specifically, the problem of how to guide the distribution of firm support

when, in the absence of it, some incumbent firms may go bankrupt and disappear.

The key dynamic concern here is that, when several firms are at stake and not all

can be supported, what particular firms are chosen may lead to subsequent effects

that are drastically different. For, as one firm is protected but an other fails, the

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latter may generate a cascade of ensuing failures that has major longer-run effects.

The question then is: what is the best criterion to use if the objective is to minimize

the overall (intertemporal) impact? In particular, we may ask: are current prices

and induced profits the right market signal? In brief, the answer we provide is that,

in some cases, profitability may by itself be a very misleading criterion and that

other network-based criteria (embodying considerations of intercentrality) will, in

general, be much more appropriate.

We end this short introduction with a brief review of related literature. From a

methodological viewpoint, our approach is close to recent work by Acemoglu et al.

(2012) who, building upon a Cobb-Douglas model proposed by Long and Plosser

(1983), study how/whether microeconomic shocks on individual agents or sectors

may aggregate into generating significant aggregate effects at the macroscopic level.

Previous papers by Horvath (1998), Gabaix (2011) are motivated by a similar con-

cern. As already explained, our objective in this paper is very different and so are

some of the key assumptions and questions asked. Thus, just to mention one signif-

icant difference, we do not make the assumption of constant returns in production

since we are interested in how profit performance is affected by network structure.

Besides, our analysis focuses on the effect of distortions, structural changes, or dif-

ferent kinds of demand and support policies rather than the spread of shocks, which

are very different kinds of phenomena.

Our work is also related to Jones (2011) who, building as well on the framework

introduced by Long and Plosser (1983), studies how misallocation at the sector

level affects GDP. More generally, our paper relates to the vast literature that

has attempted to understand the intrasectoral basis of economic development and

the sectoral policies that can mitigate either misallocation or/and coordination

problems. The workhorse in this literature has been the input-output methodology

originally formulated by Leontief (1936), which has spawned a huge body of work,

both theoretical and empirical (see e.g. the monograph Miller and Blair (2009)

for a recent account). Early on, building upon this rise of input-output analysis,

the influential work of Hirschman (1958) highlighted the importance of some of the

notions (e.g. forward and backward linkages) that will help us understand key forces

underlying our model. In contrast with the traditional input-output literature,

Hirschman’s approach emphasizes unbalanced growth rather than equilibrium as

the primary tool of dynamic analysis. To adopt such a perspective is also our

eventual objective, even though in the present paper our analysis is still mostly

static.

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However, the preliminary dynamic analysis of evolutionary market forces un-

dertaken in the last part of the paper (cf. the summary above) embodies some of

the considerations implicitly highlighted by Hirschman’s work. It also hints at the

role that network-based processes of propagation – e.g. of default through failure

cascades – should have in a proper assessment of economic policy in a complex

interconnected economy. This, in turn, leads us to the rich and diverse literature

on contagion – for example, financial contagion – that has experienced a signifi-

cant impetus in recent years (see the recent Handbook edited by Bramoulle et al.

(2016) and the recent interesting work by Baqaee (2015) on the specific context

of production networks). A different, and complementary, aspect in the dynamic

study of economic networks concerns the processes of endogenous (payoff-guided)

network formation. The network-formation literature is still in too-preliminary a

state to provide much help in this endeavor. Our analysis of how certain structural

changes on nodes and links affect agents’ payoffs is a very preliminary step in this

direction. The recent paper by Oberfield (2012) studies a stylized version of the

problem where agents endogenously select their production techniques, conceived

as links that connect them to a unique supplier.

The rest of the paper is organized as follows. First, in Section 2 we present

the benchmark model, followed in Section 3 by the introduction of a collection of

basic results. These results include, specifically, an account of how the topological

features of the production network shape the corresponding outcomes (i.e. profits

of the firms and utility of the consumer). Then, in Section 4 we study how different

kinds of distortions (uniform or firm-specific) affect the allocation of resources and

relative firm performance. In Section 5 we investigate how the global structure

of the production network determines the production and the welfare potential

of the economy and how local perturbations in the network structure impinge on

the equilibrium outcomes. Section 6 discusses the influential push-pull dichotomy

stressed by Hirschman (1958) through the lens of our model. In Section 7 we provide

a preliminary exploration of issues pertaining to firm dynamics and point out that

market signals can be quite misleading from the point of the long run welfare. We

summarize our contribution and conclude in Section 8.

All our formal proofs of our main results are included in the Appendix to the

present paper. We have also prepared two additional online appendices where we

discuss some complementary material. Specifically, in the online Appendix A we

study the general case with unrestricted heterogeneity, while in the online Appendix

B we analyse the effect of distortions for the case where the economic system is

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closed and thus monetary flows must be balanced.

2 Benchmark model

We model an economy consisting of a finite set of firms N = {1, 2, ..., n} and a

single representative consumer. Our focus, therefore, is on the interfirm production

relationships – i.e. the production network – through which the economy eventually

delivers the net amounts of consumption goods enjoyed by the consumer. In prin-

ciple, we allow that any specific good may be consumed, used as an intermediate

input in the production of some other good, or display both roles simultaneously.

The goods that provide some utility to the consumer are labeled consumption

goods, and are included in the set M ⊂ N , with m = |M | and c = (c1, c2, ..., cm)

representing a typical consumption bundle. For simplicity, leisure is not in the set

M and thus yields no utility. Thus the consumer’s endowment of labor (which is

normalized to unity) will be inelastically supplied by the consumer and hence only

has an instrumental role as a source of income. The consumer’s preferences over

consumption bundles are represented by a Cobb-Douglas utility function U(·) of

the form

U(c) =m∏i=1

cγii (1)

where each γi represents the weight that the consumer’s preferences attributes to

consumption good i.

Concerning production, we assume that there is a one-to-one correspondence

between firms and goods (thus, in particular, we rule out joint production).2 The

production of each good k takes place under decreasing returns to scale, and requires

both labor and intermediate produced inputs. The set of intermediate inputs that

firm k uses in its production is denoted by N+k , with nk = |N+

k |. Let lk stand

for the amount of labor used in the production of good k and let (zjk)j∈N+k

be the

associated amount of intermediate goods. Then the amount yk of good k produced

by the homonymous firm is determined by the production function fk : Rnk×R→ R2Conceptually, we can think of each good as a “sector” consisting of many identical firms.

Thus, from this perspective, what our model describes is the behavior of a typical firm in itscorresponding sector, all firms belonging to a given sector producing perfectly substitute goods.This, of course, is a classical way of rationalizing, and providing foundations for, competitivebehavior in equilibrium, as postulated below (see also Definition 2 in Appendix).

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which is taken to display the following Cobb-Douglas form:

yk = fk

((zjk)j∈N+

k; lk

)= Akl

βkk

∏j∈N+

k

zgjkjk

αk

(2)

where the vector (gjk)k∈N reflects the intensity (assumed positive)3 with which firm

k uses/requires its different inputs, and αk > 0 and βk > 0 are the output elasticities

of labor and intermediate inputs.4

It is convenient to view the intensities gjk of input use as reflecting relative

magnitudes, so unless mentioned otherwise we shall normalize the corresponding

vector to satisfy∑

j∈N gjk = 1 – in other words, we assume that the matrix G is

column-stochastic. On the other hand, we assume that the production technologies

exhibit decreasing returns, hence we posit that αk +βk < 1. It is worth noting that

the model allows for full heterogeneity across firms k ∈ N , not only in their pattern

of input use (gjk)k∈N but also in their production elasticities αk and βk. The latter

heterogeneity, however, does not raise particularly interesting issues and, therefore,

for the sake of formal simplicity, in the main text we shall focus throughout on

the case where αk = α and βk = β for some common values α and β. A detailed

analysis for the fully heterogeneous case may be found in the online Appendix A.

It is common in economic models to posit that more advanced technologies

employ a wider range of intermediate inputs, this being conceived as a reflection of

higher “production complexity”. Here we choose to formalize this idea in the way

suggested by Benassy (1998) (see also Acemoglu et al. (2007)), by setting

Ak = nα+νk (3)

with α + ν > 0. With this formulation, it is easy to see that a positive value

of the parameter ν corresponds to the case where a wider input range enhances

productivity, while a negative value reflects the opposite situation.5

3If gjk = 0 for some input j used in the production of a certain good k, such an input is neitheruseful nor required in k’s production. Therefore, it might as well be ignored altogether and thecorresponding link jk eliminated from the production network.

4This formulation implies that labor and all intermediate inputs in N+k are essential in pro-

duction. When the production technology of N+k = ∅, we interpret

∏j∈N+

kzgjkjk ≡ 1, thus no

production is possible because we then have Ak = 0 (see below for details).5To understand this heuristically, suppose that firm k has a total amount of money M to

spend among nk intermediate inputs used in the production of good k. Then, if each input entersproduction symmetrically and also bears the same price, firm k should split M equally among thenk inputs. Fixing the amount of labor used, this implies that production must be proportional to

nα+νk Mα(

1nk

)α= nνkM

α. Thus, if ν > 0 there are benefits from increasing the range of inputs

used in production, and the magnitude of ν quantifies precisely this effect.

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The inputs j ∈ N+k used in the production of good k can be viewed as the

in-neighbors of node k in the production directed network Γ = {N,L}, where

the set N of its vertices is identified with the set of firms and there is a directed

edge (i, j) ∈ L whenever j uses good i as an input, i.e. iff gji > 0. This is a

discrete (binary) network that only provides a qualitative account of the production

structure of the economy. A full-fledged description of this structure is provided by

the matrix of production intensities G = (gij)i,j∈N , which in turn can be regarded

as the adjacency matrix of the directed weighted network that completely describes

the production structure. The matrix G, together with the elasticity parameters α

and β, and the utility function U(·) of the representative consumer jointly provide

a full description of the economy. To study its performance we shall focus on the

standard notion of Walrasian (or Competitive) Equilibrium (WE), which consists

of a collection of prices and quantities that satisfy the usual optimality and market

clearing conditions. More precisely, a WE is an array [(p∗, w∗), (c∗,y∗,Z∗, l∗)] such

that the following conditions hold:

1. The consumption plan c∗ = (c∗1, c∗2, ..., c

∗m) maximizes U(c) subject to the

budget constraint given by the wage and profit income the consumer earns.

2. The production plans given by the outputs y = (y∗1, y∗2, ..., y

∗n), the demands for

produced inputs Z∗ = (z∗ij)i,j∈N , and the labor demands l∗ = (l∗1, l∗2, ..., l

∗n) are

all technologically feasible and maximize, for each firm i ∈ N , its respective

profit.

3. The labor market and the markets for produced goods clear (i.e. supply equals

demand).

A rigorous formalization of WE is provided by Definition 2 in Appendix. Since

the economy satisfies the usual properties contemplated by General Equilibrium

Theory, existence of a WE readily follows.

3 Basic results

The form of the production function (2) has the implication that (provided α and

β are both positive) labor and at least one intermediate input are essential in the

production of any good. No firm, therefore, can be active (i.e. achieve a positive

production) unless it relies on some other active firm. This imposes a natural

requirement of “systemic balance” on an economic system if all its firms are to be

active. And such a condition not only has static implications but, as we shall see

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in Section 7, it entails dynamic consequences as well. Similar considerations have

been found to be important in the evolution of many other systems – biological,

ecological, or chemical – where some suitable notion of systemic balance is also key

to static stability and dynamic evolution of the system.6

However, an important feature of economic systems that has no clear counterpart

in other contexts is that, in a market economy, the “source of value” is not just

internal to the production network but, crucially, is also “externally” dependent

on consumers’ preferences. Preferences provide the standard to measure welfare

and also determine the market value that shapes firm performance. To discuss this

issue, it is useful to extend the directed production network Γ = {N,L} with an

additional node c that stands for our representative consumer. This node c has an

in-link (j, c) originating in every node j ∈ M that produces a consumption good.

Such an extended network is denoted by Γ.

As a preliminary step in our analysis, we want to understand when, given a

particular (extended) production structure Γ, a particular firm i can be active at a

WE. To this end, we rely on the notion of Strongly-Connected Component (SCC).

Let the notation j � k indicate that there is a directed path originating in j and

ending in k. Then, a subset of nodes Q ⊂ N is said to be a SCC if, from every

pair of nodes j, k ∈ Q, j � k. The following result specifies necessary conditions

for some particular node to be active at a WE.

Proposition 1

Consider any firm i ∈ N that is active at some WE. Then we have:

(a) ∃Q ⊂ N that defines a SCC of the (extended) directed network Γ s.t. ∀j ∈ Q,

j � i;

(b) i� c.

Proof. See Appendix.

A simple illustration of the necessary conditions contemplated in Proposition 1

is provided by the production network shown in Figure 1. First we note that in

this network the four firms/nodes coded in red – i.e. 1 to 4 – do not satisfy the

necessary conditions (a)-(b) specified in Proposition 1 and thus cannot be active

at any WE. The violation of these conditions occur because either they do not

have any intermediate input to rely on (Firm 4) or there is no direct or indirect

connection to the consumer node (Firms 1 to 3). In the first case, the issue is one

6See, for example, the interesting work of Jain and Krishna (1998), who stress the importance of“autocatalytic balance” (the analogue of what we have called systemic balance above) in processesof growth in biological systems.

9

of feasibility (production of good 4 is not at all possible), while in the second case

the problem is one of incentives (if the goods were produced, they would fetch no

market value and hence lead to a zero profit).

In contrast, the remaining blue nodes 5-11 satisfy the aforementioned conditions

and, in principle, could be active at a WE. For nodes 5-7 this follows from the

following two-fold observation: they define a SCC (so they jointly define a feasible

production structure), and they also have an (indirect) connection to the consumer

node (hence they may contribute to market value). Instead, firms 8 to 11 do not

form part of a SCC, and thus have to depend on “external inputs” to undertake

production. They can, however, obtain those inputs from the aforementioned SCC

and then access market value through firm 11, which is connected to consumer node

c (i.e. produces a consumer good).

1

2

3

4

5

6

7

8

9

10

11

C

Figure 1: An extended production network G that includes the consumer node.The blue nodes are those that satisfy conditions (a)-(b) in Proposition 1, while thered ones do not. The green node represents the consumer demand.

Next, we address the question of whether there are conditions (possibly more

stringent than those specified in Proposition 1) that are not only necessary but

also sufficient, i.e. characterize when a firm is active at a WE. Building upon the

ideas underlying Proposition 1 and the Cobb-Douglas specification of the model,

we arrive at the following result.

Corollary 1

Consider any given firm i ∈ N . This firm is active at any WE if, and only if,

it satisfies (a) and (b) in Proposition 1 and so happens as well for all other firms

j ∈ N+i providing inputs to it.7

7Note that, clearly, if a good i satisfies (b) in Proposition 1, then all its inputs satisfy it as

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Proof. See Appendix.

In view of the previous result, throughout this paper we shall assume that all

existing firms satisfy (a)-(b). For short, this will be labeled Condition (A). Clearly,

this condition can be assumed without loss of generality, for any other firm can be

simply ignored in the analysis.

Of course, the identification of what firms are active at a WE provides only a very

partial description of the situation. In general, there will be a wide heterogeneity

in performance across firms (specifically, in terms of sales and profits). And if

firms are symmetric in every respect except for their network position, it is the

overall topology of the production network that should be used to explain such

heterogeneity. Can we map the network-performance relationship in a sharp and

insightful manner? To answer this question we turn to our first main result of the

paper, Proposition 2 below.

Proposition 2

There exists a unique WE for which the vector of equilibrium revenues s∗ = (s∗i )ni=1 ≡

(p∗i · y∗i )ni=1 is given by

s∗ =w∗(1− α)

β(I − αG)−1γ (i = 1, 2, ..., n),

while the corresponding equilibrium profits π∗ = (π∗i )ni=1 = (1− α− β)s∗.

Corollary 2

Assume M = N (i.e. all goods are consumed). Then, the relative revenues and

profits of the different firms at the WE satisfy:(s∗i∑nj=1 s

∗j

)n

i=1

=

(π∗i∑nj=1 π

∗j

)n

i=1

= (1− α)(I − αG)−1γ.

Proof. See Appendix.

Proposition 2 establishes that – under the maintained condition (A) – the unique

WE induces an equilibrium revenue (as well as profit) for each firm i that is propor-

tional to a suitable measure of network centrality of this firm given by (I−αG)−1γ.

As we explain next, this centrality notion integrates two main factors:

(i) the utility weight of each of the consumption goods whose production the

firm’s output contributes to, either directly or/and indirectly;

(ii) a weighted discounted measure of the direct and indirect ways in which the

aforementioned contribution takes place.

well. Hence the only relevant requirement concerning i’s inputs is that they be supported fromthe production side, i.e. that they satisfy (a).

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The measure of network centrality that arises from our analysis is a variation of

the widely-used concept of Bonacich centrality. Since this notion of centrality is

defined in the literature in slightly different forms, let us start our discussion by

introducing the particular version to which we are referring here.

Definition 1

Consider a directed weighted network whose adjacency (n×n)-matrix is G. Let

δ ∈ (0, 1) be a discount factor and ζ > 0 a scale factor. Then, the associated vector

of Bonacich centralities is given by v(G, δ, ζ) = ζ(I− δG)−11, where 1 is a suitable

column vector whose components are all equal to 1.

To understand intuitively the notion of Bonacich centrality,8 suppose first that

the network is binary, so that links either have weight equal to 0 or weight equal

to 1, i.e. gij ∈ {0, 1} for every i, j ∈ N . Then, since the matrix G is assumed

column-stochastic, its different columns have to include exactly one unit entry and

the other entries must be equal to zero. In this simplified case, it is easy to show that

the centrality of a particular node i can be interpreted as (is proportional to) the

average of the discounted number of paths that start at node i and reach all n nodes

(including itself) in r steps (r = 1, 2, ...). The discount factor used is δ ∈ (0, 1),

and the total discount imposed on any given path is tailored to its length, i.e. it is

δr if its length is r. To see this, rewrite the expression for centrality introduced in

Definition 1 as follows:

v(G, δ, ζ) = (vi(G, δ, ζ))ni=1 = ζ

[∞∑r=0

δrGr

]1 (4)

so that, for each i ∈ N , its corresponding centrality is given by

vi(G, δ, ζ) = ζ

n∑j=1

[∞∑r=0

δrg[r]ij

]

where each g[r]ij represents the ij-th entry of the matrix Gr. The suggested inter-

pretation is then a consequence of the fact that g[r]ij simply counts the number of

paths of exact length r that start at node i and end at node j. As we have seen (cf.

Proposition 2), the discount factor to be used in our case is δ = α (where α is the

output elasticity of intermediate inputs). Therefore, it turns out to be convenient

to have a scale factor ζ = (1−α)/n. Since the matrix G (and therefore every power

Gr) is column-stochastic, this scaling amounts to normalizing the centrality vector

v to lie in the n− 1 simplex, i.e.∑n

i=1 vi = 1.

8The measure introduced in Bonacich (1987) is defined by the expression c(G, a, b) = b(I −αG)−1G1 while Ballester et al. (2006) use instead b(G, a) = (I−αG)−11 as the measure they callBonacich centrality.

12

More generally, when the matrix G is a general column-stochastic matrix with

its entries gij ∈ [0, 1], an analogous interpretation can be provided but the “number

of paths” must then be replaced by the “normalized intensity” that flows between

pairs of nodes for every possible path length. Applied to our production context,

such intensity simply corresponds to the product of the input-demand flows between

any given firm i and the firms j that use the former’s good as input, directly or

indirectly.

The relationship between a firm’s performance and its Bonacich centrality is

especially stark in the context considered in Corollary 2. There we abstract from any

asymmetry associated with the consumption side: all produced goods are assumed

to be consumption goods and equivalent for the consumer, which implies that γ =1n1. Then, this result establishes that the relative performance of firms is exclusively

determined by their Bonacich centrality in the production network for a discount

factor δ = α and scale factor (1 − α), where we recall that α is the production

elasticity of intermediate inputs. In this sense, we may say that, given this elasticity,

the profitability of a firm is the reflection of purely “topological” features of the

production network.

In the general case where the vector of preferences γ is arbitrary and possibly

M 6= N (not all goods are necessarily consumption goods), matters must be corre-

spondingly adapted and the notion of centrality considered has to account for those

asymmetries. Then, the relevant measure of centrality associates to the paths that

arrive to any particular node j the weight γj that the consumer attributes to the

respective good in her utility function (cf. Proposition 2). Thus, in particular, if

good j is not a consumption good, paths of any given length r that connect some

node i to such a node j do not contribute directly to the centrality (and therefore

the profitability) of firm i. They may only do so indirectly to the extent that such

paths can be constituent subpaths of longer ones that eventually connect i to some

consumption good k (in which case they would be weighted by γk and affected by

a lower discount factor).

Having characterized the situation on the production side of the economy, now

we turn to studying its consumption side. Specifically, our aim is to understand how

the features of the network impinge on the consumer’s welfare at the WE. Again

we find that the vector of centralities plays a prominent role, although in this case

additional technological parameters are also important. We start the analysis by

providing an explicit expression for the equilibrium utility in our next result.

Proposition 3

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At the WE the consumer’s utility is given by

logU(c) =n∑i=1

γi log γi + log(1− α)αα

1−α +1

1− α(ν + α)∑i

vi log ni − (1− α− β)∑i∈N

vi log vi + α∑i∈N

∑j∈N+

i

vigji log gji

(5)

where recall that ni is the number of inputs used in the production of good i, ν is

the parameter modulating the economies of scope through the factor Ai = nα+νi ,

and v(G,α, 1−α) = (vi)ni=1 is the corresponding vector of Bonacich centralities for

δ = α and ζ = 1− α.

Proof. See Appendix.

The previous result highlights the three endogenous magnitudes that shape wel-

fare in our context:

1. The weighted average (log-)connectivity,∑

i vi log ni, where the degree of each

node/firm is weighted by its respective centrality.

2. A measure of heterogeneity/dispersion among firms, as reflected by the en-

tropy of their centralities, (−∑

i∈N vi log vi).

3. The average heterogeneity/entropy in input use (i.e. −∑gji log gji) displayed

by the technologies of the different firms i ∈ N , each of these individual

magnitudes weighted by the centrality of the respective firm i.

4. The heterogeneity/entropy across goods,∑n

i=1 γi log γi, displayed by the pref-

erences of the representative consumer.

Specifically, we find that consumer welfare is increasing in the average log-connectivity

of firms and in inter -firm symmetry, while it is decreasing in the intra-firm sym-

metry displayed by their technologies. These different magnitudes are aggregated

through a simple affine function whose coefficients are given by the underlying tech-

nological parameters.

Next, to obtain a sharper characterization of the situation, it is useful to reduce

the degrees of freedom by postulating the following two symmetry assumptions:

(S1) For any given firm i ∈ N , its different inputs play a symmetric role in its

production technology, i.e. gji = gki for all j, k ∈ N+i .

(S2) All goods are consumption goods and therefore γi = 1/n for every firm i ∈ N .

Under the previous conditions, the intra-firm heterogeneity of each firm i is depen-

dent on ni alone, i.e. on the number of inputs it uses (which, heuristically, could be

14

understood as the “complexity” of its production technology). Then, the expression

in (5) is substantially simplified, as stated by the following corollary.

Corollary 3

Assume (S1) and (S2) above. Then, the utility of the consumer at the WE is given

by

logU = − log n+ log (1− α)αα

1−α +1

1− α

(ν

n∑i=1

vi log ni − (1− α− β)n∑i=1

vi log vi

)(6)

Proof. See Appendix.

Thus, under input symmetry, connectivity and inter-firm entropy are the sole con-

siderations that impinge on consumer’s welfare. As we shall discuss in Subsection

5.1, this stark conclusion will also allow us to obtain a similarly sharp assessment of

what production structures are welfare optimal in different technological scenarios.

4 Price distortions

In this section, we focus on the study of what could be interpreted as taxes, dis-

tortions, or policy/price interventions of different sorts. An obvious, but still im-

portant, characteristic of an interconnected economy is that any such distortion or

intervention cannot be studied just locally. In general, it is to be expected that its

first-order impact could turn out to be quite different from its overall effect on the

economy, once the complete chain of indirect effects is taken into account. Such

a full-fledged analysis of the situation, however, can be very complex and we need

effective tools to carry out a proper analysis of the situation. Here we provide a

step in this direction.

Specifically, in this section we consider two different cases. First, in Subsection

4.1, we consider a price distortion that applies directly to all firms in the economy

in a uniform manner. Then, in Subsection 4.2, we turn to studying distortions that

apply directly to just one firm in the economy, so the effects on all others are only

indirect. In both cases, uniform or individualized, we focus on situations that, at an

abstract level, can be conceived as inducing “price wedges”. More concretely, this

is an approach that can be taken to capture a variety of different cases: government

policies that favor a particular sector, ad-valorem taxes or subsidies, or the reflection

of market power.9 To stress the intended generality, we shall simply speak of them

9See, for example, the paper by Hsieh and Klenow (2009) or Jones (2011) for an elaboration

15

as distortions10.

A common idea that arises in the analysis of both uniform and individual distor-

tions is that centrality-based measures are key in determining the size and direction

of the induced effects. Centrality being an inherently global measure, the point is

then that the impact of any change or intervention, no matter how “local” it might

appear, must be evaluated globally. In fact, variants of the same general idea will

reappear as well in much of the analysis conducted throughout the paper, again an

indication of the unavoidable global nature of the issues being studied. In essence,

the crucial network magnitudes involved in the analysis will turn out to be what we

shall call (bilateral) in- and out-centralities. These are the elements of the matrix

M = (mij)ni,j=1 ≡ (I − αG)−1 included in the definition of (Bonacich) centrality –

cf. Definition 1.

Specifically, for every pair of nodes i and j, the ij-incentrality is identified with

mij. By writing

vi(G,α, ζ) =1− αn

n∑j=1

[∞∑r=0

αrg[r]ij

]=

1− αn

n∑j=1

mij (7)

it is indeed apparent that mij represents (or, more precisely, is proportional to) the

contribution of node j to the Bonacich centrality of i. Reciprocally, we refer to the

entry mji as the ij-outcentrality. Note that, within our general economic model

(cf. Proposition 2), for each i ∈ N the entries mij (j = 1, 2, ..., n) are the weights

given to the preference weights γj of each good produced in the determination of

the equilibrium profits of firm i through the expression

π∗i = (1− α− β)w∗(1− α)

β

n∑j=1

mijγj. (8)

Thus, as explained, the contribution to i’s centrality provided by an intermediate

product j that is not consumed (whose γj = 0) has no effect on i’s equilibrium

profits.

on such a general interpretation of price distortions.10Within our framework one can rely on the same tools used here to analyze input specific

distortions, i.e. distortions that lower the marginal product of one input relative to others (see(Hsieh and Klenow, 2009) for details). An analysis of this type of distortions is not included inthe present paper, but details on it are available upon request.

16

4.1 Uniform price distortion

We start our analysis of distortionary effects by considering the implications of a

uniform price distortion τ imposed on all firms of the economy. Its effect is to draw

a proportional wedge between the price pi the consumer pays for each good i and

the price (1− τ)pi received by the firm selling it. In principle, the value of τ might

be negative, in which case it could be interpreted, for example, as a subsidy (hence

amounting to a proportional increase in the revenue earned from the firm). An issue

that arises here is whether the monetary payments (or proceeds) entailed should be

distributed back to (or subtracted from) the revenue available to the agents of the

economy – that is, whether the system is to be conceived as closed to those monetary

flows. In general, the answer must depend on the specific interpretation attributed

to those flows (e.g. on whether they are redistributive, or purely distortionary and

hence wasteful). Here, in the main text, we shall consider the formally simpler case

where they are a pure outflow (or inflow) of resources, while referring the reader to

the online Appendix B for a consideration of the alternative closed-system version.

None of our results are qualitatively affected by the scenario being considered.

Our first observation (see Lemma 1 in Appendix for details) is that, under a

uniform τ applied to all goods of the economy, the expression that characterizes the

vector of equilibrium profits is generalized to

π∗(τ) = (1− τ)(1− α− β)w∗(1− α)

β(I − α(1− τ)G)−1 γ. (9)

This readily implies that, as expected, the effect on equilibrium profits of an increase

in τ is unambiguously negative – the reason, of course, is simply that, in our context,

any distortion is always detrimental.11 Indeed, if we consider the marginal effect of

increasing τ , we find (see Lemma 2 in Appendix):

dπ∗

dτ(τ) = −(1− α− β)s∗(τ)− α(I − α(1− τ)G)−1Gπ∗(τ) < 0

where π∗(τ) stands for the full vector of profits earned under τ by all firms.

Having settled the question of how a uniform distortion affects firms’ profits,

next we turn our attention to the much more complex issue of how it can diversely

impinge on the relative profits of the different firms, as a function of their individual

position in the network. To account for this relative performance in a convenient

manner, it is useful to focus on the normalized (equilibrium) profits (πj)ni=1 of each

firm i obtained by setting the nominal wage w(τ) so that∑

j∈N πj = 1. A formal

11An analogous conclusion holds for the effect of τ on consumer’s welfare, which can be shownto be always negative. Details are available upon request.

17

characterization of the (marginal) implications of changing τ on the relative profit

performance of the different firms is provided by the following result.

Proposition 4

Given any given distortion τ ∈ [0, 1], let (πj(τ))i∈N be the corresponding profile of

normalized equilibrium profits and denote (mij(τ))ni,j=1 ≡ (I − α(1− τ)G)−1. The

marginal effects on profits induced by a change on τ are determined by the following

system of differential equations:

dπidτ

(τ) =1

1− τ

n∑j=1

mij(τ) (γj − πj(τ)) (i = 1, 2, ..., n), (10)

and hence, at a situation with no distortion (τ = 0), the marginal impact of intro-

ducing it is:

dπidτ

∣∣∣∣τ=0

=n∑j=1

mij (γj − πj(0)) (i = 1, 2, ..., n), (11)

where recall that M = (mij)ni,j=1 is the matrix of incentralities.

Proof. See Appendix.

The above result highlights that the firms most negatively affected in their rel-

ative profit performance by the introduction of the distortion are those whose cen-

trality is heavily dependent on firms with high original profits. A particularly stark

manifestation of this idea is displayed in (11), which applies to the case where the

distortion is just being marginally introduced. This expression can be heuristically

understood as follows. The effect on the relative profit standing of any given firm

changes as prescribed by an average composition/multiplication of two magnitudes:

(a) the first-order impacts experienced by each one of the firms in the economy,

whose sign and size is captured for each firm j ∈ N by (γj − πj(0)) – a

comparison of firm j’s relative profit and the (relative) weight of its output

in the consumer’s preferences;

(b) the extent to which those impacts are transmitted to the firm i in question,

as captured by its respective vector of ij-incentralities (mij)nj=1.

When the base distortion is not zero but is some given τ > 0, this two-fold

mechanism exhibits the general form given by (10), which has an analogous in-

terpretation. An interesting feature of this expression is that the sensitivity of

relative equilibrium profits to interfirm asymmetries grows steeply as τ approaches

18

unity. Another interesting consideration worth highlighting is that, because the

dependence of profits on τ is nonlinear, there is the potential for complex (and, in

particular, nonmonotonic) behavior as the distortion changes within its full range

[0, 1]. The following simple example illustrates that this is indeed a possibility.

Example 1

Consider the simple production network with nine firms depicted in Figure 2, where

it is assumed that all firms produce a consumption good and hence the arrows only

indicate the flow of input use. Figure 3 traces the relative profits12 for possible

values of τ ∈ [0, 1] and a subset of firms (for the sake of readability, not all firms

are included). The second diagram shows that, as τ grows (and, naturally, the

profit spectrum across firms narrows) there are rank changes in the relative position

of some firms. Thus, while Firm 6 remains the highest-profit firm throughout, as τ

grows the position of Firm 8 deteriorates, from second to fourth position. Indeed, we

find that it is not just that the ranking partially changes but even the relative profit

of an individual firm may evolve in nonmonotonic way as τ rises. Specifically, as

shown more clearly in Figure 4, this happens to Firm 1, whose relative profit first

grows and then decreases.

1

2

3

456

7

8

9

Figure 2: A simple production network illustrating the discussion included in Ex-ample 1.

12We fix the values of the parameters, α = 0.7, β = 0.1

19

0.0 0.2 0.4 0.6 0.8 1.0τ

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

π

Firm 1

Firm 5

Firm 6

Firm 7

Firm 8

Figure 3: The relative profits of all firms in the production network displayed inFigure 2 for all values of τ ∈ [0, 1].

0.0 0.2 0.4 0.6 0.8 1.0τ

0.111

0.112

0.113

0.114

0.115

0.116

0.117

0.118

0.119

0.120

π

Firm 1

Figure 4: The nonmonotonic behavior of the relative profit of firm 1 in the produc-tion network displayed in Figure 2, as τ changes in the range [0, 1].

4.2 Individual price distortion

In this subsection, our focus turns from a common distortion that affects uniformly

all firms to one that impinges only on a specific firm k. As shown in Appendix, such

a firm-specific price distortion, denoted by τk, introduces just a simple modification

in the equilibrium expressions derived for the benchmark model, but one that is of

a quite different nature from those obtained for a uniform distortion. For example,

in contrast with (9), the induced equilibrium profits π∗(τ ok ) = [π∗i (τk)]ni=1 are now

given by:

π∗(τk) = (1− α− β)w∗(1− α)

β

(I − αG(τk)

)−1γ. (12)

20

where G represents a modified matrix that replaces the original one, G. These

matrices only differ in their respective kth columns, gk and gk, which satisfy gk =

(1− τk)gk.

The difficulty lies in studying the inverse in (12) to obtain its dependence on τk.

To do this, it is useful to write the modified matrix G(τk) as follows:

G(τk) = G− τkgke′k, (13)

where e′k is the n-dimensional row vector that has a 1 in its kth position and 0

elsewhere. This then allows us to rely on the following result in Linear Algebra (cf.

Sherman and Morrison (1949) or Hager (1989)):

Sherman-Morrison Formula. Let A be a nonsingular n-dimensional real ma-

trix, and c, d two real n-dimensional column vectors such that 1+d′A−1c 6= 0.

Then,

(A+ cd′)−1 = A−1 − A−1cd′A−1

1 + d′A−1c.

Applying the above result to our case – with the particularization A = I − αG,

c = τkgk, and d = ek – we arrive at the following full characterization of how the

k-specific distortion affects equilibrium profits.

Proposition 5

Consider a distortion τk imposed on firm k and let (π∗i (τk))ni=1 stand for the cor-

responding equilibrium profits. The marginal effects on profits induced by a change

on τk are determined by the following system of differential equations:

dπ∗idτk

(τk) = −π∗k(0)mik

(1− τk(1−mkk))2 (14)

and hence, at a situation with no distortion (τk = 0), the marginal impact of intro-

ducing it is:

dπ∗idτk

∣∣∣∣τk=0

= −π∗k(0)mik (i = 1, 2, ..., n), (15)

Proof. See Appendix.

The previous result states that the profit decrease experienced by any firm i due

to the direct distortion impinging on another firm k depends on

• how profitable k was prior to the distortion;

• the importance of k in determining the centrality of i;

• how much of k’s centrality “feeds into” itself.

21

As intuition would suggest, while the first two considerations increase the profit loss

on firm i due to distortion on k, the last one decreases it (with the only exception

considered in (15) when there is no distortion to start with). Again we observe that

incentralities (as captured by the mik and mkk mentioned in the last two items) play

a prominent role in shaping the overall global effect. It is also interesting to observe

from (14) that the function mapping k’s distortion into i’s profit is convex so that,

just as in the uniform-distortion case, the marginal effect of increasing τk grows

with the level of it. Another feature that was noted for the previous uniform case is

that the induced nonlinearities can lead to somewhat paradoxical conclusions. This

phenomenon arises as well here, as illustrated below by through two examples.

Example 2

Consider the production network depicted in Figure 5, where the corresponding pro-

file of Bonacich centralities is also shown. Again, for simplicity, all produced goods

are assumed to be not only intermediate inputs but consumption goods as well. Firm

1 and then Firm 2 are those with the highest centrality in the production network.

Therefore, under no distortion, they are also the firms enjoying the highest profits

at the Walrasian equilibrium.

12

3

4

5

67

8

9

10

1 2 3 4 5 6 7 8 9 10

Firm

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Centr

alit

y

Figure 5: A production network in the upper panel, with the corresponding profileof Bonacich centralities for each of the nodes displayed in the lower panel.

22

However, as shown in Figure 6, the situation can change in interesting ways if

our focus turns to relative profits13 (normalized to add up to one) and the distortion

τ2 experienced by Firm 2 varies. (Note that here we are allowing τ2 to be negative,

varying in the range τ2 ∈ [−1, 1] and thus playing possibly the role of a subsidy.)

For τ2 = 0 (no distortion) the profit ranking exactly mimics that of centralities, as

already explained. In contrast, as τ2 grows and becomes positive, we observe the

somewhat paradoxical fact that the relative profit of Firm 2 monotonically grows

and, eventually, if τ2 is high enough, even surpasses that of Firm 1 and becomes the

highest. The opposite state of affairs is found if Firm 2 is subject to a negative τ2.

Then, a higher absolute value for it induces a lower profit for this firm, eventually

leading it to fall below that of Firm 10. Why does this happen? The reason is

that, as the distortion varies, the effect of this change – which is modulated by the

various incentralities – impinges most strongly on those firms i whose i2-incentrality

is highest, which then alters the relative position of Firm 2 in the direction opposite

to the change of τ2. A further illustration of this role of incentralities, which is

particularly simple and stark, is provided by our next example.

1.0 0.5 0.0 0.5 1.0τ2

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

πi

Firm 1

Firm 2

Firm 3

Firm 5

Firm 10

Figure 6: The change in relative profits among the firms placed in the productionnetwork depicted in Figure 5 as the distortion experienced by Firm 2 changes inthe range [−1, 1].

Example 3

Suppose that ten firms are arranged into a ring production network with each firm

i = 1, 2, ..., 10 using the good produced by Firm i− 1 as the sole intermediate input

and having its produced good be the sole intermediate input in the production of

Firm i + 1 (of course, indices 0 and 11 are interpreted as 10 and 1, respectively).

As in the previous examples, all produced goods are assumed to be valuable to the

13We fix the values of the parameters, α = 0.7, β = 0.1

23

consumer, and we fix α = 0.7, β = 0.1. Consider now a distortion experienced by

Firm 1, τ1 ∈ [−1, 1].

Figure 7 depicts how the relative profits of firms change over the whole range of

variation of τ1. The most affected is Firm 10, either negatively if τ1 > 0 or positively

if τ1 < 0. The reason is that this firm is the one with the highest i1-incentrality.

Following it, the firm whose bilateral incentrality relative to 1 is the highest is Firm

9, and thus this firm is the one which is the second most affected by changes in τ1,

and so on along the ring. In the end, we find that, indeed, the firm that is least

affected is Firm 1, the firm that is directly subject to the distortion. Of course,

this stark conclusion is an artifact of the extreme production network considered but

should help clarify the key mechanism underlying global effects in our context.

1.0 0.5 0.0 0.5 1.0τ1

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

πi

Firm 1

Firm 4

Firm 6

Firm 8

Firm 10

Figure 7: The change in relative profits among the firms placed in a ring productionnetwork of ten firms as the distortion experienced by Firm 1 changes in the range[−1, 1]. In such a ring network, the input links define the cycle 1→ 2→ 3→ ...→10→ 1.

5 Network structure

In line with our core concern of understanding how economic structure affects per-

formance, here we undertake an analysis of the following issues. First, in Subsection

5.1, we study what features of the production structure (e.g. whether it is more or

less connected, or its degree of heterogeneity) impinge on the production and wel-

fare possibilities attainable at equilibrium. Then, in Subsections 5.2 and 5.3, our

focus turns to studying the overall effect of single “local perturbations” affecting

the production structure. Specifically, we consider two of them: (i) the creation

(or elimination) of a link ij; (ii) the elimination of an existing node/firm, or the

24

creation of a new one. As we shall explain, these changes can be understood as

reflecting alternative economic phenomena.

5.1 Optimal network structure

The structure of production networks can be studied along many different dimen-

sions. Here we show that, as far as its impact on (consumer) welfare is concerned,

the internode symmetry displayed by the network and its connectivity stand out as

two key features. To highlight their effect, it is useful to simplify the analysis by

focusing on networks that satisfy the assumptions (S1)-(S2) introduced at the end

of Section 3. That is, we postulate that (a) the inputs involved in the production

of every good play a symmetric role, and (b) all goods are consumption goods. We

also suppose that the number of goods in the economy is given, while postponing

to Section 7 an analysis of what the welfare implications of a change in this fea-

ture of the economy are. Under these conditions, we can build upon the analysis

undertaken in Section 3 to arrive readily at the following conclusion.

Proposition 6

Assume (S.1)-(S.2) in Section 3. Then, the production structure that maximizes

consumer utility is a regular network (i.e. all firms have the same number of inputs).

Furthermore, if ν > 0 the optimal network is complete (i.e. for all i ∈ N, ni =

|N+i | = n− 1), while if ν < 0 the optimal network is a minimally connected regular

network (for all i ∈ N, ni = 1).

Proof. See Appendix.

The previous result follows from the inspection of the expression determining

equilibrium utility in (6), as a function of the the parameters of the economy and

the induced profile of firm centralities. On the one hand, the last term of that ex-

pression calls for the maximization of the entropy of the simplex-normalized vector

of centrality profiles, as given by −∑

i∈N vi log vi. This maximization is attained

when all nodes have the same centrality, which in turn requires that the network be

symmetric across nodes and hence regular. Thus all firms must display the same

“input complexity,” i.e. they should all use the same number of inputs. Then,

whether such common complexity should be maximal or minimal sharply depends

on the sign of the parameter ν in (3), which reflects the nature of the “economies

of input scope” in production. If ν > 0, complexity is beneficial and hence the

optimal production structure should be complete, i.e. all other goods should be

25

used in the production of every one of them. Instead, if ν < 0, the exact opposite

applies and the optimal production structure should display the minimum complex-

ity consistent with viability (cf. Corollary 2). That is, it should constitute a ring

for example.

In principle, of course, it is far-fetched to entertain the possibility that the pro-

duction structure of the economy might be “designed.” Instead, it is more natural

to conceive this structure to be a reflection of a wide number of different forces (e.g.

technological change or any other modification of the underlying economic envi-

ronment) whose magnitude and direction can hardly be controlled. Therefore, the

discussion in this subsection has to be interpreted mostly as a conceptual exercise.

That is, the objective is to gain some understanding of what features of the produc-

tion structure of an economy have an important impact on welfare. And, from this

perspective, what Proposition 6 shows is that (abstracting from some other consid-

erations) two characteristics arise as key: symmetry across all production processes

and extreme exploitation/avoidance of all economies/diseconomies of scope.

5.2 Link creation

In the remaining part of this section, we adopt a local perspective to the analysis

of changes in the production structure. First, in this subsection, we focus on the

impact of a change affecting a single link – for concreteness, we consider the case

where the link is created. A preliminary and immediate point to note is that if a

link ij is formed, the firm i that consequently sees an expansion in the uses of its

output cannot have its relative profits decrease. The reason is that its centrality

cannot decrease by this change – clearly, the set of paths that access consumer

nodes can only grow.

The effect of the new link ij on other firms– in particular, the firm j whose

range of inputs has increased – is in general ambiguous (see Example 4 below).

It depends, for example, on how the weights (gkj)k∈N+j

are affected by the change.

Among the different possibilities one could contemplate in this respect, here we shall

assume, for concreteness, that the new matrix G continues to be column stochastic.

A possible interpretation of this modeling choice is that the new link entails a new

and full-fledged production technology available to firm j that, if adopted by this

firm, satisfies the same normalization condition as all other technologies do. In a

sense, we view it as embodying a mature new option that can be selected by firm

26

j as a coherent and balanced package.14

Proposition 7

Consider an initial production network with adjacency matrix G and suppose that

a link ij is added to it. Denote by G the resulting adjacency matrix and let q =

(qk)nk=1 ∈ Rn (

∑k∈N qk = 0) be the real vector that reflects the entailed adjustment

across the two adjacency matrices, i.e. their respective jth columns, (gkj)nk=1 and

(gkj)nk=1, satisfy gkj = gkj + qk for all k = 1, 2, ..., n with gij = qi > 0. The change

on equilibrium profits, (∆π∗k)nk=1, induced by the new link is given by:

∆π∗k = απ∗jmki qi +

∑`∈N+

jmk` q`

1 + qimji + α∑

`∈N+jmj` q`

, (16)

where π∗j is the original profit of firm j under adjacency matrix G.

Proof. See Appendix.

The previous result establishes that if a new link ij is added to the production

network, the effect on the equilibrium profit of any particular firm j is given by the

composition of two key factors:

• The profit originally obtained by the firm j that provides a new use to good

i. Intuitively, this magnitude reflects the importance of firm j in providing

the immediate market value generated by the new link ij.

• The impact on the centrality of k of each of the inputs involved in the produc-

tion of j (including the new input i), as captured by the respective incentrali-

ties, mki and (mk`)`∈N+j

. Each of these incentralities is weighted by the factor

q` that reflects the adjustment made (positive or negative) when transforming

the adjacency matrix G into G.

Again, therefore, we can understand the changes resulting from the new link as

consisting of the composition of two kinds of effects: a first-order effect whose mag-

nitude is associated to the equilibrium-induced importance of the directly affected

nodes; a second-order effect that involves the transmission of the aforementioned

first-order effect through the full array of network-based interactions that are cap-

tured by the matrix of incentralities.

14Admittedly, this is only one of the possibilities that could be reasonably considered. Alterna-tively, it could be assumed that the components of the new (gkj)k∈N+

jsum up to some positive

magnitude H 6= 1. Then, one could still unit-normalize the corresponding j-th column of G byadjusting the overall input elasticity of firm j to the value αj = Hαj . None of these possiblemodeling variations would affect the essential gist of our analysis.

27

Example 4

As advanced, in this example we illustrate that the addition of a new link ij can

deteriorate the relative position of the firm whose input range is expanded by the new

link. Consider the production networks depicted in Figure 8, and suppose that the

smaller network gives rise to the larger one that includes the new link 1 → 4. Let

us assume that all goods are consumer goods and have an equal weight in the utility

function of the consumer. Let us also suppose that the new matrix G displayed

after the change continues being column stochastic. The respective profits15 for the

original and modified networks are displayed next to the nodes in Figure 8.

1

π1*=0.015

2

π2*=0.066

3

π3*=0.061

4

π4*=0.058

1

π1*=0.034

2

π2*=0.057

3

π3*=0.055

4

π4*=0.054

Figure 8: Two different production networks, whose only difference concerns theexistence, or not, of the link from node 1 to node 4. All nodes/firms are supposedto produce a consumption good with equal preference weights.

Thus one finds that firm 4 obtains a lower profit after link 1 → 4 is added to

the original network. Why is this the case? The simple reason is that, as explained,

centrality (and hence profitability) is associated to the (discounted) value a product

delivers to the consumer, not to the value it allows other firms to attain. That is,

profitability is gathered downstream rather that upstream and, given the architec-

ture of the network in our example, the discounted downstream “flow” gained by

Firm 4 from the new link falls relative to that of the other firms – in particular,

that of Firm 1, whose centrality is sure to increase.

5.3 Node deletion

In this section, we study the alternative (local) change in the production structure

given by the addition or removal of a (single) node. Now, for concreteness, we focus

our attention on the case where the change involves the removal of the node. This

is, in fact, the kind of change that will be considered in Section 7 when studying the

chain implications of a process of node/firm bankruptcy and endogenous removal.

15We fix the values of the parameters, α = 0.7, β = 0.1, and normalize w∗ = β

28

Once more we must face the modeling question of how the setup is to be renor-

malized when the network changes – in this case, the set of nodes. Now it concerns

both the utility and the production functions, all of which deliver a value of zero

when any of its arguments is equal to zero. For the utility function, the renormal-

ization we choose to tackle the problem is the natural one: we restrict its arguments

to the range of consumer goods which can actually be produced given the prevailing

set of active firms. In Subsection 7.1, we provide a specific motivation for this choice

in a context where the number of firms changes due to a process of firm selection.

Concerning the renormalization to be implemented on the production functions

when there is a reduction in the inputs available, again for concreteness we choose

one specific option but other alternatives could be analogously considered. The

alternative considered is different from that adopted in Subsection 5.2, where the

change involved an increase in input use. As we now explain, this contrast is

motivated by a different view on what is the nature of the phenomenon underlying

the situation in either case. Here we conceive the disappearance of a node as a

somewhat abrupt/unexpected event that requires some short-run and hence partial

adjustment. Thus, unlike what was suggested for the case of link creation, we

assume that the columns of the matrix G are not renormalized. Thus, firm j that

see one of its inputs vanish somehow adapts (for otherwise the indispensability of

all prior inputs would force it to a zero production) but in a partial manner. Along

with the point made in Footnote 14, we can interpret this as a downward shift in the

productivity in the overall use of the available intermediate inputs.16 Admittedly,

one should not expect real-world situations to be so markedly polar as suggested

between the increase and decrease in the use of preexisting inputs. The contrast,

however, is plausible and we find it useful to simplify the discussion of either case.

Before stating the main result from this subsection, let us introduce the following

notation. Let G denote the resulting adjacency matrix after the deletion of node

i. If the deleted firm produced a consumption good, then after deletion of that

node one may expect that weights γ that the consumer’s preferences attribute to

all remaining goods will adjust17. Let γ denote the vector of resulting preference

weights after a node is deleted.

In view of the relationship established between profitability and centrality, it is

quite clear that the elimination of a node can only be detrimental to the profits of

16If we proceeded in this way, a full-fledged analysis of the situation should rely on the extendedframework considered in the online Appendix A, where full firm heterogeneity is allowed – inparticular, on input elasticities.

17We discuss this adjustment in more details in 7.1.2

29

all firms in the economy. A precise specification of the relative magnitudes of the

effects that apply to each of them is given by the following result.

Proposition 8

Consider an initial production network with adjacency matrix G and suppose that

node i ∈ N = {1, 2, ..., n} is removed from it. Denote by(∆π∗j

)j 6=i the change in

equilibrium profits of the remaining firms. Then,

∆π∗j (G) = πj(G)− π∗j (G)− πi(G)mji(G)

mii(G), (17)

where π(G) = (1− α− β)1−αβw∗(I − αG)−1γ

Proof. See Appendix.

The previous result indicates that the effect induced by the removal of a partic-

ular firm i on the profits of the remaining firms j 6= i exhibits the usual pattern:

a first-order effect (that is quantified by the prominence/profitability of the firm

directly affected, i.e. firm i) composed with the effect that the directly affected

firm has on the centrality of any other firm (measured by the corresponding incen-

trality). In the present case, however, the incentrality terms mji are “normalized”

by the effect mii that firm i has on its own centrality – in this sense, the relevant

magnitude scales the outcentralities of i by the extent to which this firm’s cen-

trality effect feeds into itself. The additional effect, which may have an opposite

sign, is due to potential preference adjustments and is captured by the difference

πj(G) − π∗j (G). Note that when there is no adjustment in consumer’s preferences

(when γj = γj ∀j ∈ {1, 2, .., n} \ {i}) equation (17) becomes

∆π∗j (G) = −π∗i (G)mji(G)

mii(G).

6 Forward and backward linkages

As explained in the Introduction, one of the useful features of the model is that it

allows a transparent formalization of the notions of forward and backward linkages

(also called push and pull effects) that, as stressed by (Hirschman, 1958), can be

important in assessing, for example, public interventions and economic policies. In

essence, all of the comparative static results considered so far in Subsections 4.1

through 5.3 can be conceived as involving those two types of linkages, forward and

backward. That is, the overall effects considered so far can be seen to include

a forward linkage that operates downstream on the production structure, and a

30

backwards one that works upstream.

Rather that revisiting all our previous results in this light, it will prove more

useful at this point to discuss these forces/linkages in the context of two particularly

clear-cut instances: a (Hicks-neutral) pattern of technological improvements on

the production technologies of the different firms, and an arbitrary change in the

preferences of the consumer across the different goods. As we shall see, while the

first case exerts only pure “push” forces on the equilibrium outcome, the second

one embodies an array of “pull” forces (some positive and others negative).

6.1 Technological change

Consider a change in the pre-factors (Ak)k∈N of the production functions of each

firm k in the economy (cf. (2)-(3)), which become (Ak)k∈N . We shall refer to

them as the productivities of the respective firms. The key observation to make

from Proposition 2 is that the equilibrium conditions imply that the induced sales

at equilibrium are independent of those productivities. This readily leads to the

following two conclusions:

(i) The equilibrium input demands [(z∗jk)j 6=k]k∈N by each firm k (which are pro-

portional to sales) remain unaltered by the change in productivities. Thus, in

this sense, there are no pull effects operating upstream along the production

structure.

(ii) The equilibrium prices of the goods enjoying higher productivities decrease

proportionally no less than their respective productivities rise. This in turn

triggers a downstream push over all sectors that use those goods directly or

indirectly, with lower prices all along inducing more output being produced.

Item (i) is quite clear and requires no further elaboration. It is worth mentioning,

however, that for the same reason why input demands are unaffected by the change,

profits are unaffected as well. Productivity changes, therefore, have no effect on the

inter-firm distribution of profits. The reason why this happens is highlighted in

Item (ii). In the new equilibrium, price changes adapt to absorb all quantity ef-

fects (direct and indirect) resulting from increased productivities. This, to be sure,

is crucially dependent on our Cobb-Douglas formulation, which implies that price

changes exactly absorb all quantity effects (direct and indirect) resulting from in-

creased productivities. Admittedly, such a consequence of the postulated functional

form is quite extreme, but has the advantage of highlighting the contrast between

31

forward and backward linkages in a clear-cut manner. A precise account of how

these price-mediated effects propagate upstream is provided by the following result.

Proposition 9

Consider a set of changes in firm productivities such that the new levels, (Ai)i∈N ,

are related to the former ones as follows:

Ai = ξiAi (ξi > 0, i = 1, 2, ..., n). (18)

Under a suitable normalization, the corresponding equilibrium prices (p∗i )i∈N and

(pi)∗i∈N , respectively prevailing before and after the change, satisfy:

− (log p∗i − log p∗i )ni=1 = −(I − αG′)−1 (log ξi)

ni=1

or

p∗ip∗i

=n∏k=1

ξ−mkik (i = 1, 2, ..., n), (19)

where G′stands for the transpose of the adjacency matrix of the production network

and, for each firm i, (mki)k∈N is the vector of its outcentralities.

The key implication of the previous result is that, as explained in (ii), the inter-

firm linkages in this case are mediated through prices and quantities alone (i.e. not

revenues nor profits) and operate downstream. That is, the first-order effect of the

change in the productivity of any given firm k, as embodied by its corresponding ξk,

propagates through the outcentralities of this firm. Concerning prices, this effects

is as described in (9). On the other hand, in view of the constancy of equilibrium

revenues, the corresponding effect on quantities is simply given by

y∗iy∗i

=n∏k=1

ξmkik (i = 1, 2, ..., n). (20)

6.2 Preference changes

Here we illustrate the nature of the backward linkages by focusing on a characteristic

example: changes in preferences, as captured by variations in the preference weight

vector γ – cf. (1). Recall that, as established by Proposition 2, equilibrium revenues

and profits are given by

s∗ = (s∗i )ni=1 ≡ (p∗i · y∗i )

ni=1 =

w(1− α)

β(I − αG)−1γ (i = 1, 2, ..., n),

and

π∗ = (π∗i )ni=1 = (1− α− β)s∗.

32

Thus both magnitudes, revenues and profits, are determined by linear functions of

the preference weights γ = (γi)i∈N . This readily leads to the following conclusion,

which we state formally for the sake of completeness.

Proposition 10

Suppose that the preference weights change from γ to γ, with∑

j∈N ∆γi ≡∑

j∈N γi−γi = 0. Then, the induced change in equilibrium profits, (∆π∗i )i∈N , satisfies:

∆π∗i ∝∑j∈N

mij ∆γj (i = 1, 2, ..., n). (21)

Proof. Omitted.

We find, therefore, that in contrast with the distribution-invariant forward link-

ages induced by changes in firms’ productivities, any variation in preferences not

only has consequences on the allocation of resources but also distributional impli-

cations on firms’ profits. Specifically, we have that the change in the profit of any

given firm i is proportional to a weighted average of the changes (positive or nega-

tive) experienced by the relative importance that the consumer’s utility attributes

to each consumption good. Naturally, for each firm i, the weight that a particu-

lar consumption good j has in the computation of such an average effect is given

by the incentrality mij that captures the direct and indirect impact of good j on

the centrality of firm i. The overall effect on the whole set of firms is purely re-

distributive, in the sense that the total profits are unaffected by the changes in γ.

The following corollary – which is an immediate consequence of the fact that the

matrix of incentralities M = (mij)ni,j=1 ≡ (I − αG)−1 is column stochastic – states

it explicitly.

Corollary 4

Under the conditions specified in Proposition 10, any change in the preference

weights γ induces a change in equilibrium profits, (∆π∗i )i∈N , that satisfies Σi∈N∆π∗i =

0.

7 Firm dynamics

To address in this section the issue of firm dynamics, we start by introducing in

Subsection 7.1 a theoretical framework that extends the basic setup considered in

Section 2 along two complementary directions. First, we incorporate fixed labor

costs. This has the main effect of allowing for the possibility that, even at equilib-

rium, a firm may incur losses. If this occurs, we assume it means that the firm in

33

question becomes bankrupt and thus may disappear. The second extension consid-

ers the case where the number of goods in the economy may vary, thus leading to

the possibility the size of the production network may change.

In Subsection 7.2 we combine the two aforementioned extensions to provide a

preliminary analysis of firm dynamics. More specifically, our discussion focuses

on comparing the welfare implications of alternative mechanisms governing the

disappearance of firms when, say, exogenous shocks may threaten their survival.

The main insight we gather through a simple illustrative exercise can be summarized

as follows: if one relies on strict market-based indicators (e.g. profits) instead

of other network-based criteria, the consequences from the viewpoint of long-run

welfare can be quite detrimental.

7.1 A generalized framework: fixed costs and varying net-

work size

7.1.1 Fixed costs

Suppose that, as long as any given firm i remains active, it has to pay some fixed cost

fi ≥ 0, which is interpreted as some given labor requirements that are independent

on the scale of the production. This induces two different changes in our basic

framework. First, given the wage w, prices (pj)nj=1 for the intermediate inputs, and

the production plan [yi, li, (zji)nj=1)], the profit of firm i is given by

πi = piyi −n∑j=1

pjzji − w(li + fi).

Second, while the demand functions for labor and intermediate inputs do not change

with the introduction of fixed costs, the market-clearing condition for the labor

market does change since, in this case, the amount of labor available for production

decreases with the number of active firms. Specifically, the new market-clearing

condition of the labor market becomes:∑li = 1− F (22)

where F ≡∑n

i=1 fi stands for the aggregate fixed cost across all firms.

The former modifications on the theoretical framework in turn induce some

changes on the equilibrium analysis of the economy. First, as a counterpart of

original Proposition 2, we have the following generalization:

Proposition 11

34

There exists a unique WE for which the vector of equilibrium revenues s∗ = (s∗i )ni=1 ≡

(p∗i · y∗i )ni=1 is given by

s∗ =w∗(1− F )(1− α)

β(I − αG)−1γ (i = 1, 2, ..., n),

while the corresponding equilibrium profits are π∗ = (π∗i )ni=1 = (1− α− β)s∗−w∗f,

where f = (f1, f2, ..., fn)′.

Proof. See Appendix. Arguably, in a market environment such as the one con-

sidered here, the profit earned by a firm is one of the benchmarks we would like

to contemplate in measuring its performance. Then, quite naturally, the require-

ment that no negative profits (i.e. no losses) be incurred by a firm arises as well

as a natural benchmark to use in assessing its “viability.” Under the simplifying

assumption that all goods are consumption goods, the implications of such a profit-

based criterion are formally spelled out in the following straightforward but useful

corollary.

Corollary 5

In the context considered in Proposition 11, the individual firm profits obtained at

the WE satisfy:

π∗i ≥ 0 ⇐⇒ vi ≡ (1− α)n∑j=1

mijγj ≥β

1− α− βfi

1− F(23)

where (mij)nj=1 is the ith row of the matrix of incentralities M (cf. (7)).

Proof. Omitted.

The above Corollary indicates that, in order for a firm i to be viable, its network

(in-)centrality must be no lower than a certain given threshold, whose magnitude

only depends on technological/cost parameters, the size of the economy, and the

fixed costs. In Subsection 7.2, we compare this criterion with another one that

also relies on the global architecture of the network but changes focus from how

profitable a firm is to how much it impinges on the profits of other firms – that

is, from how central a firm is to how it affects the centrality of others. This, in

the end, boils down to shifting attention from aggregate incentrality to normalized

outcentrality, as suggested by Proposition 8. Indeed, this is the approach also

pursued in the present context by the following result, which specifies the impact

that the disappearance of any given firm has on the profits of all others.

Proposition 12

Consider an initial production network with adjacency matrix G and suppose that

35

node i ∈ N = {1, 2, ..., n} is removed from it. Denote by(∆π∗j

)j 6=i the change in

equilibrium profits of the remaining firms. Then,

∆π∗j = πj(G)− π∗j (G)− πi(G)mji

mii

(j =, 1, 2, ..., n; j 6= i), (24)

where π(G) = (1− α− β) (1−α)w∗(1−F+fi)β

(I − αG)−1γ

Proof. See Appendix.

An interesting observation that follows from the previous result is that, in con-

trast with the case with no fixed costs, the removal of a firm i could now conceivably

have a positive influence on the profits of other firms j 6= i (and, conceivably, even

lead to positive overall effects). This occurs because the elimination of a firm re-

laxes the labor feasibility constraint when there are fixed labor costs. The extent

to which this is important for a particular firm j depends on the scales at which

firms i and j operated (or, equivalently, their original profits), as well as on the

the market pressure that the fixed costs fi were previously imposing on the labor

market.

7.1.2 Network size

Now we focus on the modeling issue of how to measure (consumer) welfare across

situations where the network size changes due to variations in the number of firms

active in the economy. To this end, we need a formulation of preferences that

is of course consistent with our Cobb-Douglas specification (1) but is also able

to accommodate a varying number of consumption goods. A useful route to do

so is provided by the so-called Ethier-Dixit-Stiglitz (EDS) preferences, which are

represented by the utility function:

U(c ; ρ,N) =∑i∈N

cρi (25)

for some ρ > 0, where N is the whole universe of all possible goods (for simplicity,

all assumed to be valued by the consumer). An equivalent representation of the

same preferences is embodied by the monotone transformation of U(·) that induces

a CES-format counterpart given by:

U(c ; ρ,N) =

[∑i∈N

cρi

] 1ρ

.

Clearly, given any ρ, the above formulation can be adapted to any nonempty subset

of goods M ⊂ N by simply changing the set of goods under consideration. More-

36

over, as it is well-known, the corresponding function U(· ; ρ,M) converges to the

Cobb-Douglas (CD) utility function (1) with equal weights γi = 1m

in the limit of

ρ→ 0. It is in this sense that, for any given set M of consumption goods, we may

view our CD formulation as a representation of EDS preferences with an elasticity

of substitution 11−ρ converging to 1.

The former considerations indicate that one can suitably compare the welfare

of the consumer for different sets of goods being consumed, say M and M ′, by

resorting to the functions U(· ; ρ,M) and U(· ; ρ,M ′), defined respectively as direct

generalizations of (25). Then, again taking the limit on ρ for both of them, we arrive

at the conclusion that

limρ→0+

U(c(ρ) ; ρ,M) = |M | ≡ m; limρ→0+

U(c′(ρ) ; ρ,M ′) = |M ′| ≡ m′ (26)

where c(ρ) ∈ Rm and c′(ρ) ∈ Rm′ stand for the pair of equilibrium consumption

vectors associated to ρ and, respectively, M and M ′. To understand (26), simply

note that, under (25), individual optimality requires that the equilibrium consump-

tion of all goods be positively bounded away from zero in ρ. Thus, in the end, we

conclude that the welfare comparison of two equilibrium allocations with different

sets of available consumption goods, M and M ′, comes down, in the Cobb-Douglas

limit, to a comparison of their respective cardinalities, m and m′. This will be,

therefore, the welfare criterion used in our ensuing discussion.

7.2 Firm dynamics: an illustrative example

We now rely on the formalization and results of the previous subsection to undertake

a preliminary exploration of firm dynamics. We focus, specifically, on the effects

of alternative firm-selection criteria and, to fix ideas, consider the simple example

with 6 firms depicted in Figure 9.

12

3

4 5

6

Figure 9: Initial production network. It consists of 6 nodes, with firms {1, 2, 3} and{4, 5, 6} defining two completely connected cliques.

37

To understand the role played by each node i in the network, a natural way to

proceed is to focus on its bilateral incentralities and outcentralities. On the one

hand, we know from Proposition 11 that, if the preference vector γ is symmetric,

the equilibrium profit of a firm i, is proportional to its aggregate incentrality mIi ≡∑

j∈N mij (cf. (7)). This magnitude, therefore, can be seen as a market-based

measure of firm i’s own performance at equilibrium. In contrast, the aggregate

outcentrality mOi ≡

∑j∈N mji is a measure of how important firm i is for the

aggregate (in-)centrality of all other firms. In fact, Proposition 12 shows that if the

latter magnitude is normalized by mii (i’s contribution to its own centrality), the

induced adjusted outcentrality, mOi ≡ 1

mii

∑j∈N mji, is an important component in

the aggregate effect of i’s removal on all other firms. This in turn suggests that, in

order to anticipate the final impact of the removal of any given firm i, one should

compare mIi and mO

i .

To apply these considerations to our example, in Figure 10 we depict the profiles[mIi

]i∈N and

[mOi

]i∈N across all 6 nodes in the production network being consid-

ered (for expositional clarity we normalize the wage by setting w = 1). The key

observation to make is that there is an acute contrast between the incentralities and

adjusted outcentralities across nodes, with a polar behavior when one compares the

sets {1, 2, 3} and {4, 5, 6}.

1 2 3 4 5 6Node

5

10

15

Value

Adjusted out-centrality

In-centrality

Figure 10: Graphical representation of the incentrality profile and the profile ofadjusted outcentrality for the 6 firms of the economy, α = 0.9

In view of the aforementioned contrast, the question arises as to what criterion

(in- or out-centrality) should be used to evaluate the “systemic importance” of

the different firms in our example. This question, as posed, is probably too vague

to be really useful. Thus let us frame it in the following more concrete scenario,

somewhat artificial but also transparent and clear-cut. Suppose that a common

temporary shock hits two firms, say Firm 1 and 6, which in the absence of any

outside support are sure to go bankrupt and disappear from the economy. Further

38

assume that the funds that may be used to provide such a support are limited in

that only one of the two firms can be helped. Which of the two should it be?

To provide all necessary information, suppose that vector of fixed costs f satisfies

f1 = f6 = 10−8 and f2 = f3 = f4 = f5 = 0.0235. Then, it can be computed that the

equilibrium profit is given by π∗ = (0.278, 0.255, 0.255, 9 × 10−5, 9 × 10−5, 0.025)′.

Thus, as our theory prescribes, the three first firms with the highest centralities

earn the highest profits, while the last three display the lowest profit levels.

Naturally, the answer to the question we have posed must depend on what the

objective function to be optimized is. In line with the discussion undertaken in

Subsection 7.1.2, let us suppose that the final objective is to maintain the largest

possible number of active firms in the economy. Then, one possible criterion to

determine what firm should be saved from bankruptcy is given by profitability, i.e.

what we have labeled the “market-based measure of performance.” If one relies on

this criterion, the scarce funds should be used to help Firm 1 and hence it would

be this firm that survives while Firm 6 would go under. However, once this has

been carried out – interpreted as a once and for all intervention – the market forces

continue being at work and can therefore have a further impact on the survival

of the remaining firms. In fact, we find that Firms 4 and 5 now incur losses in

the Walrasian Equilibrium (WE) of the smaller economy with the set of active

firms given by {1, 2, 3, 4, 5}, and hence those two firms should subsequently exit the

market. Thereafter, all the firms in the remaining set {1, 2, 3} avoid losses at the

corresponding WE and hence the situation may remain stationary in the absence

of any further interference or a subsequent entry of new firms. This process is

graphically illustrated in Figure 11.

(a) (b) (c)

1

π1*=0.28

2

π2*=0.26

3

π3*=0.26

4

π4*=9×10-5

5

π5*=9×10-5

6

π6*=0.02

1

π1*=0.29

2

π2*=0.26

3

π3*=0.26

4

π4*=-1×10-4

5

π5*=-1×10-4

1

π1*=0.32

2

π2*=0.29

3

π3*=0.29

Figure 11: The firm dynamics triggered by the initial elimination of Firm 6 (a),followed by the consecutive step (b) where Firms 4 and 5 incur losses (i.e. obtainnegative profits) and are eliminated. In the resulting network (c) all firms makepositive net profit

39

Alternatively, the criterion for selecting the firm to be supported may single

out the firm that, if it were to disappear, would have the highest effect on the

profit performance of other firms. Then, quantifying such an effect by the adjusted

(aggregate) outcentrality, the support to buffer the shock should be enjoyed by Firm

6 rather than Firm 1. Of course, once this choice is implemented, the immediate

consequence is that Firm 1 exits the system. The adjustment process ends there.

For, in the resulting situation, where the other five firms are still active, all of

these enjoy positive profits at the induced Walrasian Equilibrium. This process is

graphically illustrated in Figure 12. The system, therefore, reaches a stationary

state that dominates (in the sense explained in Subsection 7.1.2) the alternative

state that would have been reached by the alternative selection criterion based on

firm profitability. This contrast illustrates in a stark manner what has been the

leading theme of this paper, namely, that a networked production economy cannot

be properly understood, nor suitable policies implemented, unless one adopts a

genuine network viewpoint.

(a) (b)

1

π1*=0.28

2

π2*=0.26

3

π3*=0.26

4

π4*=9×10-5

5

π5*=9×10-5

6

π6*=0.02

2

π2*=0.38

3

π3*=0.38

4

π4*=0.009

5

π5*=0.009

6

π6*=0.033

Figure 12: The elimination of Firm 1 (a) does not cause thereafter the exit of anyother firm from the network.

8 Summary and conclusions

In this paper, we have studied a general equilibrium model of an economy that fo-

cuses primarily on the inter-firm network of input-output relationships that underlie

its production structure. Is this modeling approach useful to understand economic

performance? A first, and particularly sharp, illustration of its usefulness derives

from the connection we have identified between market equilibrium outcomes and

an intuitive measure of centrality that combines topological and preference infor-

mation. Specifically, we have shown that, at equilibrium, the profits of the different

40

firms are directly proportional to their corresponding centralities.

The paper has then turned to extending this approach into the investigation of

a number of important comparative-statics questions:

(a) the effects of distortions (local or global);

(b) the implications of alternative network changes (on nodes or links);

(c) the impact of non-network “fundamentals” (preferences or productivities).

While the specifics of each of these cases are quite different, the conclusions obtained

exhibit a common format. The overall effect is always obtained from a composition

of a first-order impact on the firms directly affected and a diffusion of this impact

throughout the economy as determined by the matrix of cross-firm inter-centralities.

A different, but complementary, perspective is gained if we decompose matters into

the effects that spread upstream along the network (backward linkages) and those

that do it downstream (forward linkages). This alternative perspective has been

found useful, for example, when comparing the changes that affect productivities

and those that impinge on preferences.

Finally, we have turned to studying the dynamic consequences of incorporating

network considerations into the evaluation of simple policy dilemmas – in partic-

ular, we have compared the implications of alternative criteria for firm support in

the face of exogenous shocks that threaten firm survival. Relying on a simple ex-

ample, we have illustrated the point that tailoring those supporting decisions only

to current market-based information (thus ignoring forward-looking network-based

considerations) may be quite misleading and lead to disappointing results. For ex-

ample, the number of firms eventually affected by the shock may be much larger

than would have been optimal.

Clearly, the model proposed in this paper is quite stylized and hence leaves am-

ple room for extensions. Interesting possibilities would be to allow for less stringent

assumptions on the competitive behavior of agents, or generalize the Cobb-Douglas

formulation of preferences and technologies posited here. In fact, a theoretical

framework with the aforementioned features has been formulated in the paper by

Baqaee (2015) already mentioned in the Introduction. Specifically, that paper con-

siders a context where firms are involved in monopolistic competition, and both

preferences and technologies are of the CES type. (On the other hand, it also al-

lows for adjustments in the extensive margin, through the endogenous entry and

exit of firms.) Baqaee’s concern, however, is similar to that of Acemoglu et al.

(2012) and thus focuses his analysis on how microeconomic shocks aggregate at the

level of the whole economy. An extension of our analysis to the richer scenario stud-

41

ied in Baqaee (2015) would be very interesting indeed, and is one of our objectives

for future research.

More generally, a primary objective for follow-up research should be to apply

our network-based approach to inform and shape a variety of measures of economic

policy. But to this end, of course, a thorough empirical testing of the model must

be conducted first. This, in turn, requires the use of data that are rich and granular

enough at the microeconomic level to account for the intricate pattern of inter-

firm interactions that characterize a modern economy. Fortunately, these data are

now becoming available, as well as the tools (theoretical, algorithmic, and com-

putational) needed to analyze them. As a case in point, we refer to the work in

progress that we are currently undertaking with panel data from the Spanish Tax

Agency. The data include essentially all bilateral transactions among Spanish firms

(and many other economically relevant entities, public or private) during the decade

2004-2013. The analysis conducted so far, still preliminary, has delivered promising

results. In particular, we have found a very strong and significant relationship be-

tween the profitability of firms and their respective centrality, which is one of the

basic predictions of our model (cf. Proposition 2). Subsequent research will test

some of its other predictions.

42

Appendix: Proofs of the main results

We start this Appendix by providing a formal definition of Walrasian Equilibrium

employed in the paper. Then we proceed by providing detailed proofs of the results

from the paper.

Definition 2 (Walrasian Equilibrium)

A WE is an array [(p∗, w∗), (c∗,y∗,Z∗, l∗)] that satisfies the following conditions:

1. The representative consumer chooses c∗ by solving:

maxc

∏i inM

cγi

s.t.∑i∈M

pici = w +n∑k=1

πk.

2. Firms choose y∗,Z∗, l∗ by solving:

max(zji)j ,li

piyi −∑j∈r+i

pjzjigji − wli

s.t. zi = Ailβi

∏j∈N+

i

zgjiji

α

.

3. Markets for labour and intermediate goods clear:

yi =∑j

zij + ci ∀(i ∈ N)∑i

li = 1.

Proof of Proposition 1: The production function (2) implies that, in order to

be active, a firm has to have at least one active in-neighbour (see footnote 4).

Consider an upstream walk from i visiting only active firms. As the number of

firms is finite and each active firm must have at least one active in-neighbor, this

means that there must exist an upstream walk from i that visits the same active

node more than once. This proves part (a).

To prove part (b), note that, in the end, the demand for produced goods in the

model is generated by the consumer. Thus if there is no path from a firm to the

consumer then the total demand for that good will be zero, and therefore that firm

will choose not to produce.

Proof of Corollary 1. Follows from Proposition 1

43

Proof of Proposition 2. To simplify notation, throughout the proofs in the Ap-

pendices we shall dispense with the asterisk to identify equilibrium magnitudes.

The context should make clear when we refer to equilibrium values. The first order

conditions for the maximization problem of firm i with respect to zji are given by

Aipiαgjilβi z

αgji−1ji

∏k,k 6=j

zαgkiki = pj ⇒ zji =piαgjipj

yi (j = 1, 2, ..., n), (27)

and with respect to labor li:

Aipiβlβ−1i

∏k

zαgkiki = w ⇒ li =piβ

wyi. (28)

Using (27) and (28) we may write the profit of firm i as follows:

πi = piyi −∑j

(pjpiαgjipj

yi

)− wpiβ

wyi = (1− α− β)piyi. (29)

Next, from the clearing condition for the labor market and (28), we find that w =

β∑

i piyi, which together with (29) and the FOC for the consumer gives rise to:

ci =

∑nj=1(1− α− β)pjyj + w

piγi =

(1− α)w

βpiγi. (30)

Then, if (27) and (30) are inserted into the market clearing condition for good i,

this condition can be written in the following manner:

si =(1− α)w

βmγi + α

∑j

gijsj

where si ≡ piyi is the revenue of firm i. In vector notation, we may compactly write

the system of market clearing conditions for all intermediate goods as follows:

s =(1− α)w

βγ + αGs (31)

which, as the matrix G is column-stochastic and α < 1, has the unique solution:

s =(1− α)w

β(I − αG)−1 γ.

Finally, also note that (29) directly implies that equilibrium profits satisfy π =

(1− α− β)s. This completes the proof.

Proof of Corollary 2. Suppose that all goods are consumed and define:

v ≡ β

ws = (1− α)(I − αG)−1γ. (32)

Expanding (32) we can write v = (1−α)(∑∞

k=0 αkGk

)γ. This sum converges since

0 ≤ α < 1 and G is column-stochastic. The latter also implies that 1′Gγ = 1,

44

as∑n

i=1 γi = 1. Given that the product of two stochastic matrices is a stochastic

matrix, the same conclusion applies if instead of G we consider Gk for any k ∈ N ,

i.e. we also have 1′Gkγ = 1. This then implies:

1′v = (1− α)∞∑k=0

αk =1− α1− α

= 1.

From the previous expression, together with (28) and the labor-market clearing

condition, it directly follows that vj =sj∑ni=1 si

. Thus, recalling that πi = (1−α−β)si,

the proof is complete.

Proof of Proposition 3. To calculate the utility of the consumer we proceed as

follows. Inserting (27) and (28) into the production function (2), we have:

yi = Ai

(piβ

wyi

)β∏j

(piαgjipj

yi

)αgji⇒

∏j∈N+

i

(pαgjij

)yi = Ai

(piβ

wyi

)β∏j

(piαgjiyi)αgji ⇒

∏j∈N+

ipαgjij

pisi = Ai

(β

wsi

)β∏j

(αgjisi)αgji .

Then, taking natural logs we get that:

α∑j∈Ni

gji log pj − log pi = (33)

logAi + β log β − (1− α− β) log si − β logw + α logα + α∑j

gji log gji (i = 1, 2, ..., n)

Let us define u ≡ α logα+ β log β + (1− α− β) log β and write the system (33) in

vector notation as follows:

α (I − αG′) logp = (1−α)1 logw−(α+ν) logn+(1−α−β) log v−αH ′1−u1 (34)

where we have used (3) and (32) to expand Ai and si respectively. Then, premul-

tiplying (34) with v′ = (1− α)γ ′(I − αG′)−1 and normalizing∑

i log pi = 0 we get

(35)

(1− α) logw = (α + ν)∑i

vi log ni − (1− α− β)∑i

vi log vi + α∑i

∑j

vigji log gji + u,

(35)

Recall now that ci = (1−α)wβpi

γi. So we can write the utility function of the represen-

45

tative consumer as follows:

U(c) =n∏i=1

((1− α)w

βpiγi

)γi=

n∏i=1

γγii1− αβ

w.

Finally, substituting w from (35) we can express the logarithm of the utility function

as

logU(c) =n∑i=1

γi log γi + log(1− α)αα

1−α +1

1− α(ν + α)∑i

vi log ni − (1− α− β)∑i∈N

vi log vi + α∑i∈N

∑j∈N+

i

vigji log gji

.

Proof of Corollary 3. In the symmetric case gji = gki = 1/ni ∀(j, k ∈ N+i ) and

γi = 1n∀i. Hence, we have that

∑i

∑j vigji log gji =

∑i vi log 1

ni= −

∑i vi log ni,

and∑n

i=1 γi log γi = − log n , which together with Proposition 3 gives:

logU(c) = −logn+ log (1− α)αα

1−α +1

1− α

(ν

n∑i=1

vi log ni − (1− α− β)n∑i=1

vi log vi

).

For the sake of completeness we now extend the definition of WE to a context

with distortions, as studied in Section 4.

Definition 3 (WE with distortions)

For an exogeneus vector of price distortions τ = (τi)ni=1. A WE is an array

[(p∗, w∗), (c∗,y∗,Z∗, l∗)] such that:

1. The representative consumer chooses c∗ to solve:

maxc

∏i inM

cγi

s.t.∑i∈M

pici = w +n∑k=1

πk.

2. Firms choose y∗,Z∗, l∗ to solve:

max(zji)j ,li

(1− τi)piyi −∑j∈r+i

pjzjigji − wli

s.t. zi = Ailβi

∏j∈N+

i

zgjiji

α

.

46

3. Markets for labor and intermediate goods clear:

y∗i =∑j

z∗ij + c∗i (i = 1, 2, ..., n)∑i

li = 1.

Next, we state and prove two separate Lemmas that establish corresponding

claims made in Section 4.

Lemma 1

There exists a unique WE with uniform price distortions as long as (1 − τ)α < 1.

The vector of equilibrium revenues s∗(τ) = (s∗i (τ))ni=1 ≡ (p∗i · y∗i )ni=1 is given by

s∗(τ) =w∗(1− α)

β(I − α(1− τ)G)−1γ (i = 1, 2, ..., n),

while the corresponding equilibrium profits π∗(τ) = (π∗i )ni=1 = (1−τ)(1−α−β)s∗(τ).

Proof of Lemma 1: The proof is analogous to that of Proposition 2. In the case

of a uniform price distortion τ , the former equations (27) and (28) become:

zji = (1− τ)piαgjipj

yi

li = (1− τ)piβ

wyi

for each i, j = 1, 2, ..., n. Therefore, proceeding as before, it readily follows that the

vector of equilibrium revenues satisfies

s(τ) =(1− α)w

β(I − α(1− τ)G)−1 γ

and, correspondingly, the equilibrium profit of each firm i is given by:

πi(τ) = (1−τ)piyi(τ)−∑j∈N+

i

pj(1−τ)piαgjipj

yi(τ)−w(1−τ)piβ

wyi(τ) = (1−τ)(1−α−β)si(τ)

or in vectorial form:

π(τ) = (1− τ)(1− α− β)s(τ) (36)

Finally, the uniqueness of equilibrium follows simply from the fact that G is column-

stochastic and (1− τ)α < 1. This completes the proof.

Lemma 2

In the WE with uniform price distortion τ the following holds:

dπ∗

dτ(τ) < 0

47

Proof of Lemma 2: Recall that equilibrium revenues satisfy s(τ) = 1−αβγ+α(1−

τ)Gs(τ). Thus, differentiating both sides with respect to τ , we get:

ds

dτ(τ) = −αGs(τ) + α(1− τ)G

ds

dτ(τ)⇒ ds

dτ(τ)− α(I − α(1− τ)G)−1Gs(τ) (37)

From (36) and (37), it easily follows that:dπ

dτ(τ) = −(1− α− β)s(τ)− α(1− τ)(1− α− β)(I − α(1− τ)G)−1Gs(τ)

= −(1− α− β)s(τ)− α(I − α(1− τ)G)−1Gπ(τ) < 0.

Proof of Proposition 4: Recall, from (36), that

π(τ) = (1− τ)(1− α− β)s(τ) = (1− τ)(1− α− β)1− αβ

w (I − α(1− τ)G)−1 γ

Thus, setting w = 1−(1−τ)α(1−τ)(1−α)(1−α−β)β, the induced profits are normalized so that

the sum of profits across nodes is equal to 1. We can now write the normalized

equilibrium profit vector as (38).

π(τ) = (1− α(1− τ))γ + α(1− τ)Gπ(τ) (38)

Taking the derivative of (38) with respect to τ we obtain:

dπ

dτ(τ) = αγ − αGπ(τ) + α(1− τ)G

dπ

dτ(τ),

and then an explicit expression for dπdτ

(τ)

dπ

dτ(τ) = α(I − α(1− τ)G)−1(γ −Gπ(τ)) (39)

Equation (39) can be developed as follows:

dπ

dτ(τ) = α(I − α(1− τ)G)−1γ − 1

1− τ((I − α(1− τ)G)−1π(τ)− (1− α(1− τ)(I − α(1− τ)G)−1γ

)=

α

1− α(1− τ)π(τ)− 1

1− τ((I − α(1− τ)G)−1π(τ)− π(τ)

)=

1

1− τ

(1

1− α(1− τ)π(τ)− (I − α(1− τ)G)−1π(τ)

)=

1

1− τ(I − α(1− τ)G)−1 (γ − π(τ))

For a particular firm i we have:

dπidτ

(τ) =1

1− τ

n∑j=1

mij(τ) (γj − πj(τ)) (40)

48

If all goods are symmetric consumption goods, (40) becomes:

dπidτ

(τ) =1

1− τ

n∑j=1

mij(τ)

(1

n− πj(τ)

)so that, when evaluated at τ = 0, we find:

dπidτ

∣∣∣∣τ=0

=n∑j=1

mij (γj − πj(0))

as desired, thus completing the proof.

Lemma 3

Let G be a n-dimensional matrix, and α ∈ R such that there exist (I − αG)−1.

Then: (I − αG)−1G = 1α

((I − αG)−1 − I)

Proof of Lemma 3: (I−αG)−1G =(∑∞

k=0 αkGk

)G =

∑∞k=0 α

kGk+1 = 1α

∑∞k=0 α

k+1Gk+1 =1α

∑∞k=1 α

kGk = 1α

(∑∞k=0 α

kGk − α0G0)

= 1α

((I − αG)−1 − I)

Proof of Proposition 5: Proceeding in a way analogous to that pursued for a

uniform price distortion, we arrive to the following expression for the revenue vector

at equilibrium.

s(τk) =1− αβ

w (I − (αG− ατkgke′k))−1γ

Using the Sherman-Morrison formula we get (41).

s(τk) =1− αβ

w

((I − αG)−1γ − (I − αG)−1ατkgke

′k(I − αG)−1

1 + ατke′k(I − αG)−1gkγ

)(41)

Note first that 1−αβw(I−αG)−1γ = s(0) and (1−α)w

β(I−αG)−1ατkgke

′

k(I−αG)−1γ =

−ατksk(0)(I − αG)−1gk. Using this to simplify (41) we get:

s(τk) = s(0)− sk(0)ατk(I − αG)−1gk

1 + ατke′k(I − αG)−1gk= s(0)− sk(0)

τk((I − αG)−1 − I)ek1− τk + τkmkk

where the last equality follows directly from Lemma 3. For a specific firm i 6= k we

have πi(τk) = (1− α− β)(si(0)− sk(0) τkmki

1−τ+τmkk

), and hence

dπidτk

(τk) = −(1− α− β)sk(0)mki

(1− τk(1−mkk))2

On the other hand, for firm k, πk(τk) = (1−τk)(1−α−β)(sk(0)− sk(0) τk(mkk−1)

1−τ+τmkk

)=

(1− τk)(1− α− β)sk(0) 1−τk1−τk+τkmkk

, and therefore

dπkdτk

(τk) = −(1− α− β)sk(0)mkk

(1− τk(1−mkk))2

49

It is then obvious that

dπi(τk)

dτk

∣∣∣∣τk=0

= −πk(0)mki (i = 1, 2, ..., n),

which completes the proof.

Proof of Proposition 6: For a fixed n = |N |, let us find the network topology

that maximizes expression (6). In order to do that it is useful to define Φ : Rn ×Rn → R with Φ(v,n) =

∑ni=1 vi log ni. Let us also define function Ψ : R→ R as:

Ψ(y) = max(xi)ni=1,(yi)

ni=1

n∑i=1

xilogyi

s.t.n∑i=1

yi = y

n∑i=1

xi = 1

yi ≤ n− 1 ∧ yi ≥ 1 ∧ xi > 0, i = 1, ..., n

One can easily show that function Ψ is strictly increasing in y. For any graph with n

nodes and L links (∑

i ni = L) it is clear that Φ(vi, ni) ≤ Ψ(L) < Ψ(n(n−1)). Note

that for complete network ∀(i, j ∈ N)(ni = nj) ⇒ Φ(vvi , nci) = Ψ(n(n − 1)) (since

Ψ is strictly increasing). Thus, the complete network is the unique maximizer of Φ.

The value of expression−∑n

i vi log vi which is in fact the entropy measure of simplex

vector v, will be maximized when vi = vj ∀(i, j ∈ N), which will incidentally be

true in the case of the complete network and the ring network. It follows now that

utility will be maximized at the complete network when ν > 0 and that the complete

network will be the unique network that maximizes the consumer’s utility. It is easy

to see that when ν < 0 a network in which all nodes have the same centrality v

and the same in-degree (equal to 1) will maximize (6), as for such a network the

entropy of centrality vector will be maximized and η∑n

i=1 vi log ni = 0. This will,

for instance, be the case when the production network is the ring network. When

ν = 0, then any network such that the centrality of each node is equal will maximize

social welfare.

Proof of Proposition 7: Using the Sherman-Morrison formula we write:

s =1− αβ

w(I − αG)−1γ =1− αβ

w(I − αG− αqe′i)−1γ

=1− αβ

w

((I − αG)−1γ +

(I − αG)−1αqe′i(I − αG)−1

1− αe′i(I − αG)−1qγ

)Proceeding analogue to the analysis in Proposition 5, we get that the effect of adding

50

a link (i, j) on the centrality of firm k is:

sk − sk = αsjqimki +

∑l∈N+

jqlmkl

1 + qimji + α∑

l∈N+jqlmjl

.

where sk is the revenue of k after adding link (i, j). Since, as we have shown before,

π = (1− α− β)s this completes the proof.

The following important lemma is proven by (Ballester et al., 2006)

Lemma 4 (BZC)

mji(G)mik(G) = mii(G)(mjk(G)−mjk(G−i))

Proof of Lemma 4:

mii(G)(mjk(G)−mjk(G−i)) =∞∑p=1

αp∑r+s=pr≥0,s≥1

g[r]ji (g

[s]jk − g

[s]j(−i)k) =

∞∑p=1

αp∑r+s=pr≥0,s≥2

g[r]ii g

[s]j(i)k

=∞∑p=1

αp∑

r′+s′=pr′≥1,s′≥1

g[r′]ji g

[s′]ik = mji(G)mik(G)

where G−i is the resulting network after the elimination of node i from network

G.

Proof of Proposition 8: We can write the profit of firm j after elimination of

node i from the network as

πj(G) = (1− α− β)1− αβ

wn∑k=1k 6=i

mjk(G)γk

= (1− α− β)1− αmii(G)β

wn∑k=1

(mii(G)mjk(G)−mji(G)mik(G)) γk.

where the second line follows directly from Lemma 4. Furthermore:

πj(G)− πj(G) =

(1− α− β)1− αβ

w

(1

mii(G)

n∑k=1

(mii(G)mjk(G)−mji(G)mik(G))γk −n∑k=1

mjk(G)γk

)=

(1− α− β)1− αβ

w

(n∑k=1

mjk(G)(γk − γk)−mji(G)

mii(G)

n∑k=1

mik(G)γk

)=

πj(G)− πj(G)− πi(G)mji(G)

mii(G)

51

Proof of Proposition 9: From (33) by normalizing: α logα+β log β−β logw =

0 and by writing bi ≡ −(1− α− β) log si + α∑n

j=1 gji log gji we get:

logp = (I − αG′)−1 (logA+ b)

Consider a technology shock that changes Ai to Ai ≡ ξiAi (we shall use ˜ to denote

the variables after the shock). It is clear that this shock will not affect the relative

demand for goods in the equilibrium, which implies that (31) will not be affected

by the technology shock. So, si = si ⇒ bi = bi. From here it directly follows:

log p− logp = (I − αG′)−1(

log A− logA)

= (I − αG′)−1 log ξ,

where we have used the following notation: logx ≡ (log xi)ni=1.

Proof of Proposition 11: Proceeding analogously to Proposition 2 one can see

that the profit of a firm at the equilibrium is πi = (1− α − β)si − wfi. Using this

and the market clearing condition for labor (22) we get that the total income of the

consumer is: Y = w+∑n

i=1 πi = w+(1−α−β)∑n

i=1 si−∑n

i=1wfi = (1−α)1−Fβw.

Then, the revenue vector in the equilibrium is defined with:

s =(1− α)(1− F )

βwγ + αGs⇒ s =

(1− α)(1− F )

βw(I − αG)−1γ

and the corresponding vector of equilibrium profits is π = (1− α− β)s− wf

Proof to the Corollary 5: Directly follows from the Proposition 11 and (32).

Proof of Proposition 12: We can write the profit of firm j after i is eliminated

from the network as:

πj(G) = (1− α− β)(1− α)(1− F + fi)

β

n∑k=1k 6=i

mjk(G)γk

= (1− α− β)(1− α)(1− F + fi)

mii(G)βw

n∑k=1

(mii(G)mjk(G)−mji(G)mik(G)) γk.

52

where the second line follows directly from Lemma 4. Furthermore:

πj(G)− πj(G) =

(1− α− β)(1− α)(1− F + fi)

βw(

1

mii(G)

n∑k=1

(mii(G)mjk(G)−mji(G)mik(G))γk −1− F

1− F + fi

n∑k=1

mjk(G)γk

)=

(1− α− β)(1− α)(1− F + fi)

βw

(n∑k=1

mjk(G)

(γk −

1− F1− F + fi

γk

)− mji(G)

mii(G)

n∑k=1

mik(G)γk

)=

πj(G)− πj(G)− πi(G)mji(G)

mii(G)

53

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